Percent error measures how far an experimental (measured) value is from the true (theoretical) value, expressed as a percentage.

Formula: Percent Error = |(Experimental − Theoretical) ÷ Theoretical| × 100

Example: Experimental = 47g, Theoretical = 50g |(47 − 50) ÷ 50| × 100 = |−0.06| × 100 = 6% error

The Percent Error Formula: Complete Breakdown

Percent Error = |(Experimental Value − Theoretical Value) ÷ Theoretical Value| × 100

The vertical bars | | denote absolute value — meaning you always take the positive result of the division, regardless of whether your experimental value was above or below the theoretical value. (The sign question is addressed separately below.)

Breaking down each component:

Experimental Value (also called the measured value, observed value, or approximate value): The result you actually obtained from your experiment, measurement, instrument reading, or calculation. This is what you recorded in the lab.

Theoretical Value (also called the true value, accepted value, exact value, or reference value): The established, known, or expected value. In chemistry and physics, this is often taken from a textbook, periodic table, or physical constant. In engineering, it is the design specification. In business, it is the forecasted figure.

The subtraction (Experimental − Theoretical): Gives you the raw error — how far off your measurement was, with a sign indicating direction (positive if you overestimated, negative if you underestimated).

Dividing by the Theoretical Value: Normalizes the error. A 3-gram error means something very different when the true value is 5g (60% error) versus when the true value is 500g (0.6% error). Division by the theoretical value makes the error meaningful regardless of scale.

The absolute value | |: Removes the sign so you get a magnitude of error, not a direction. Most academic applications use absolute value to report "how wrong" without specifying direction.

Multiplying by 100: Converts the decimal ratio to a percentage for easier interpretation and communication.

Related Error Formulas

Absolute Error

Absolute Error = |Experimental Value − Theoretical Value|

Absolute error is the simplest error measure — just the magnitude of the difference between your measurement and the true value, without any normalization. It is expressed in the same units as the original measurement.

Example: True value = 50g, Measured = 47g Absolute Error = |47 − 50| = 3 grams

Absolute error tells you how far off you were in real units. It does not tell you whether that error is large or small relative to the quantity being measured.

Relative Error

Relative Error = |Experimental Value − Theoretical Value| ÷ Theoretical Value

Relative error is percent error expressed as a decimal rather than a percentage. Multiply by 100 to convert relative error to percent error.

Example: True = 50g, Measured = 47g Relative Error = |47 − 50| ÷ 50 = 3 ÷ 50 = 0.06 (or 6% as percent error)

Relative error is the preferred form in scientific literature and data analysis because it is dimensionless and scale-independent.

Percent Error vs Percent Change

These two formulas look similar but mean very different things:

Percent Error: Compares a measured value to a known true value. The denominator is always the theoretical/true value. Used to assess measurement accuracy.

Percent Change: Compares a new value to an old value over time. The denominator is the original (old) value. Used to track change, growth, or decline.

Feature	Percent Error	Percent Change
Formula	|(Exp − Theo) ÷ Theo| × 100	((New − Old) ÷ Old) × 100
Denominator	True/Theoretical value	Starting/Original value
Purpose	Measure accuracy	Measure change over time
Sign	Usually absolute (positive)	Can be positive or negative
Used in	Science, engineering, lab work	Finance, economics, statistics

For percent change calculations, use our Percentage Calculator, which handles all four types of percentage problems including percent change, percent increase, and percent decrease.

Step-by-Step: How to Calculate Percent Error

Step 1: Identify your experimental value Write down the value you measured, observed, or calculated from your experiment.

Step 2: Identify your theoretical value Write down the accepted, true, or expected value from your reference source (textbook, data table, specification sheet).

Step 3: Subtract theoretical from experimental Experimental − Theoretical = raw error (can be positive or negative)

Step 4: Divide by the theoretical value Raw error ÷ Theoretical value = relative error (decimal)

Step 5: Take the absolute value Remove any negative sign — you want the magnitude.

Step 6: Multiply by 100 Convert to percentage.

Worked Example: Density of Iron

A student measures the density of an iron sample and gets 7.63 g/cm³. The accepted density of iron is 7.874 g/cm³.

• Step 1: Experimental = 7.63 g/cm³
• Step 2: Theoretical = 7.874 g/cm³
• Step 3: 7.63 − 7.874 = −0.244
• Step 4: −0.244 ÷ 7.874 = −0.031 (approximately)
• Step 5: |−0.031| = 0.031
• Step 6: 0.031 × 100 = 3.1% error

The student's measurement was 3.1% below the accepted value for iron density.

Practical Examples Across Multiple Subjects

Chemistry Example 1: Molar Mass Determination

A chemistry student experimentally determines the molar mass of magnesium chloride (MgCl₂) to be 93.8 g/mol. The accepted molar mass from the periodic table is 95.21 g/mol.

• Experimental = 93.8 g/mol
• Theoretical = 95.21 g/mol
• |93.8 − 95.21| ÷ 95.21 × 100
• |−1.41| ÷ 95.21 × 100
• 1.41 ÷ 95.21 × 100
• = 1.48% error

Assessment: A 1.48% error in a molar mass determination is generally considered acceptable in a high school or undergraduate lab setting, indicating good experimental technique with minor measurement uncertainty.

Chemistry Example 2: Titration

A student performs an acid-base titration and calculates the concentration of a sodium hydroxide solution to be 0.098 mol/L. The known concentration is 0.1 mol/L.

• Experimental = 0.098 mol/L
• Theoretical = 0.1 mol/L
• |0.098 − 0.1| ÷ 0.1 × 100
• 0.002 ÷ 0.1 × 100
• = 2.0% error

Assessment: A 2% error in a titration experiment is within acceptable range for a student lab, where volumetric glassware precision, indicator endpoint judgment, and burette reading all contribute to measurement uncertainty.

Physics Example 1: Acceleration Due to Gravity

A physics student uses a pendulum to measure the acceleration due to gravity and obtains 9.4 m/s². The accepted value is 9.81 m/s².

• Experimental = 9.4 m/s²
• Theoretical = 9.81 m/s²
• |9.4 − 9.81| ÷ 9.81 × 100
• 0.41 ÷ 9.81 × 100
• = 4.18% error

Assessment: A 4.18% error in a pendulum experiment is typical at the high school level, where timing precision, string length measurement, and amplitude assumptions all introduce error.

Physics Example 2: Ohm's Law Verification

A student measures the resistance of a resistor using a voltmeter and ammeter, calculating R = 47.3 Ω. The resistor's rated value is 47 Ω (±5% tolerance).

• Experimental = 47.3 Ω
• Theoretical = 47 Ω
• |47.3 − 47| ÷ 47 × 100
• 0.3 ÷ 47 × 100
• = 0.638% error

Assessment: Less than 1% error — excellent precision for this type of measurement. The result falls well within the resistor's 5% manufacturing tolerance.

Real-Life Example 1: Manufacturing Quality Control

A pharmaceutical company manufactures 500mg ibuprofen tablets. Quality control testing measures a sample tablet at 487mg.

• Experimental = 487 mg
• Theoretical = 500 mg
• |487 − 500| ÷ 500 × 100
• 13 ÷ 500 × 100
• = 2.6% error

Assessment: Most pharmaceutical standards require dosage to be within ±5% of labeled amount. At 2.6%, this tablet meets specification — but multiple samples would be tested and averaged for a full quality control assessment.

Real-Life Example 2: Business Forecasting

A retail company forecasts Q3 revenue at $2.5 million. Actual Q3 revenue comes in at $2.34 million.

• Experimental (actual) = $2.34 million
• Theoretical (forecast) = $2.5 million
• |2.34 − 2.5| ÷ 2.5 × 100
• 0.16 ÷ 2.5 × 100
• = 6.4% error

Assessment: A 6.4% forecast error is within the range most businesses consider acceptable for quarterly revenue projections, where market conditions introduce inherent unpredictability.

Example Calculations Reference Table

Experimental Value	Theoretical Value	Absolute Error	Percent Error
47 g	50 g	3 g	6.00%
9.4 m/s²	9.81 m/s²	0.41 m/s²	4.18%
93.8 g/mol	95.21 g/mol	1.41 g/mol	1.48%
47.3 Ω	47 Ω	0.3 Ω	0.638%
487 mg	500 mg	13 mg	2.60%
7.63 g/cm³	7.874 g/cm³	0.244 g/cm³	3.10%
98.5°C	100°C	1.5°C	1.50%
299,000 km/s	299,792 km/s	792 km/s	0.264%
0.098 mol/L	0.1 mol/L	0.002 mol/L	2.00%
24.5 N	25 N	0.5 N	2.00%

Error Type Reference Table

Error Type	Formula	What It Measures	Units
Absolute Error	|Exp − Theo|	Raw magnitude of error	Same as measurement
Relative Error	|Exp − Theo| ÷ Theo	Proportional error (decimal)	Dimensionless
Percent Error	|Exp − Theo| ÷ Theo × 100	Proportional error (percentage)	%
Percent Change	(New − Old) ÷ Old × 100	Change over time	%
Standard Error	SD ÷ √n	Precision of a mean estimate	Same as measurement

What Is a Good Percent Error? Quick Reference

Percent Error	Accuracy Level	Typical Context
< 1%	Excellent	Professional lab, precision instrument
1% – 5%	Good	University lab, quality manufacturing
5% – 10%	Acceptable	High school lab, standard measurement
10% – 20%	Fair	Rough estimation, field measurement
> 20%	Poor	Significant error — review technique

Important: What constitutes an "acceptable" percent error is always context-dependent. A 5% error in a rough field measurement is excellent; a 5% error in pharmaceutical dosing is a regulatory concern. Always assess percent error relative to the standards of your specific application.

Can Percent Error Be Negative?

This is one of the most frequently asked questions about percent error — and the answer requires careful qualification.

The standard formula uses absolute value, so the result is always non-negative (zero or positive).

Using |Experimental − Theoretical| ÷ Theoretical × 100, the absolute value signs ensure you always get a positive percentage, regardless of whether your experimental value was higher or lower than the theoretical value.

However — some applications report signed percent error (without absolute value):

Signed Percent Error = (Experimental − Theoretical) ÷ Theoretical × 100

In this version:

• A positive percent error means your experimental value was higher than the theoretical value (overestimate)
• A negative percent error means your experimental value was lower than the theoretical value (underestimate)

When signed percent error is used:

• When the direction of error matters (systematic vs random error analysis)
• In engineering and manufacturing to distinguish between over-specification and under-specification
• In business forecasting to distinguish between optimistic and pessimistic forecasts
• In advanced scientific papers that analyze bias direction

When absolute percent error is used:

• Most high school and undergraduate lab reports
• General accuracy assessment
• When only the magnitude of error matters, not its direction

Bottom line: In standard academic use, percent error is always positive because absolute value is applied. If your teacher or professor specifies "signed percent error," report the sign. Otherwise, report the absolute (positive) value.

Subject-Specific Applications

Percent Error in Chemistry

Chemistry is the most common context where students encounter percent error, primarily in quantitative laboratory work.

Lab contexts where percent error appears in chemistry:

Stoichiometry experiments: When a student synthesizes a compound and measures the actual yield versus the theoretical yield calculated from stoichiometry. A percent yield of 95% means 5% of the product was lost — but note that percent yield uses a slightly different convention (no absolute value, since yield is always less than or equal to theoretical).

Physical property measurement: Measuring the melting point, boiling point, density, or refractive index of a pure substance and comparing to the accepted tabulated value.

Titration and volumetric analysis: Calculating the concentration of an unknown solution and comparing to a known standard.

Atomic mass and molar mass experiments: Determining molar mass through colligative properties or gas law experiments and comparing to periodic table values.

Spectroscopy: Comparing experimentally measured absorption wavelengths to accepted values.

Common sources of error in chemistry labs:

• Instrument calibration (balance, burette, pipette, pH meter)
• Temperature effects on volume measurements
• Impure reagents or contamination
• Procedural errors (incorrect endpoint detection in titration)
• Evaporation of volatile liquids
• Parallax errors in reading graduated cylinders

Percent Error in Physics

In physics, percent error is used to evaluate experimental results against established physical constants and theoretical predictions.

Lab contexts where percent error appears in physics:

Mechanics: Measuring acceleration, velocity, force, or momentum and comparing to theoretical predictions from Newton's Laws or kinematic equations.

Waves and optics: Measuring wavelength, frequency, or refractive index and comparing to known values.

Electricity: Verifying Ohm's Law, Kirchhoff's Laws, or measuring capacitance and inductance against rated values.

Thermal physics: Measuring specific heat capacity of metals or liquids and comparing to standard thermodynamic tables.

Gravitation: Measuring g (acceleration due to gravity) using pendulums, free fall, or spring systems.

Commonly accepted physical constants for reference:

Physical Constant	Accepted Value
Acceleration due to gravity (g)	9.80665 m/s²
Speed of light (c)	299,792,458 m/s
Planck's constant (h)	6.626 × 10⁻³⁴ J·s
Avogadro's number (Nₐ)	6.022 × 10²³ mol⁻¹
Charge of electron (e)	1.602 × 10⁻¹⁹ C
Universal gas constant (R)	8.314 J/(mol·K)

Common sources of error in physics labs:

• Reaction time in manual timing experiments
• Friction and air resistance in mechanics experiments
• Instrument resolution limits (ruler, stopwatch, voltmeter)
• Environmental interference (magnetic fields, temperature variation)
• Alignment errors in optical experiments

Percent Error in Mathematics and Statistics

In pure mathematics, percent error arises when comparing approximate values (rounded, estimated, or truncated) to exact values.

Examples in mathematics:

Pi approximation: If a student uses 3.14 as an approximation for π:

• Experimental = 3.14, Theoretical = 3.14159...
• |3.14 − 3.14159| ÷ 3.14159 × 100 = 0.051% error

Using 22/7 as π:

• |3.14286 − 3.14159| ÷ 3.14159 × 100 = 0.040% error

Square root approximation: Using 1.41 for √2:

• |1.41 − 1.41421| ÷ 1.41421 × 100 = 0.298% error

In statistics, the Standard Deviation Calculator and Average Calculator work alongside percent error to give a complete picture of measurement variability and central tendency.

How to Calculate Percent Error in Excel

Microsoft Excel and Google Sheets make percent error calculation straightforward with simple formulas.

Excel formula for percent error:

Assuming:

• Cell A1 contains your Experimental Value
• Cell B1 contains your Theoretical Value

Enter in cell C1: =ABS((A1-B1)/B1)*100

This formula uses ABS() for absolute value, subtracts the theoretical from experimental, divides by theoretical, and multiplies by 100.

To format the result as a percentage with 2 decimal places: Format cell C1 as "Number" with 2 decimal places, or use: =TEXT(ABS((A1-B1)/B1)*100,"0.00")&"%"

Building a percent error table in Excel:

Column A	Column B	Column C
Experimental	Theoretical	=ABS((A2-B2)/B2)*100
47	50	6.00%
9.4	9.81	4.18%
93.8	95.21	1.48%

Copy the formula in column C down for as many rows as you need. Excel will automatically adjust the cell references.

For signed percent error (showing positive/negative direction): =(A1-B1)/B1*100 Remove the ABS() function to retain the sign.

For a dedicated scientific calculation environment, our Scientific Calculator handles full-precision arithmetic including absolute value operations.

Advanced Concepts: Significant Figures and Error Propagation

Significant Figures in Percent Error

Percent error results should be reported to an appropriate number of significant figures — typically matching the precision of your least precise measurement.

Rules for significant figures in percent error:

• If your measurements have 3 significant figures, report percent error to 3 significant figures (e.g., 4.18%, not 4.182%)
• If your measurements are rough estimates, 2 significant figures is usually sufficient (e.g., 4.2%)
• In most high school lab reports, 1–3 decimal places are appropriate

Example: Experimental = 9.4 m/s² (2 sig figs), Theoretical = 9.81 m/s² (3 sig figs) Report percent error to 2 significant figures: 4.2% (not 4.18%)

Error Propagation

When an experimental result is calculated from multiple measured quantities, errors from each measurement combine. This is called error propagation.

For addition and subtraction: Add the absolute errors. If Z = A + B, then ΔZ = ΔA + ΔB

For multiplication and division: Add the relative errors. If Z = A × B, then ΔZ/Z = ΔA/A + ΔB/B

Example: Calculating density from mass and volume measurements

• Mass: 47.3g ± 0.1g (relative error: 0.211%)
• Volume: 6.2 cm³ ± 0.1 cm³ (relative error: 1.613%)
• Density = mass ÷ volume = 7.63 g/cm³
• Total relative error = 0.211% + 1.613% = 1.824%
• Absolute error in density = 7.63 × 0.01824 = 0.139 g/cm³
• Report density as: 7.63 ± 0.14 g/cm³

Types of Experimental Error

Understanding where error comes from is as important as calculating its magnitude:

Random Error: Unpredictable fluctuations that cause measurements to scatter around the true value. Random error can be reduced by taking multiple measurements and averaging. It affects precision.

Systematic Error: Consistent, directional errors caused by instrument calibration issues, flawed procedures, or environmental factors. Systematic error cannot be reduced by repeating measurements — it must be identified and corrected. It affects accuracy.

Human Error: Mistakes in reading instruments, recording data, or following procedures. Unlike random and systematic error, human error is not inherent in the measurement process — it is avoidable.

Instrument Limitation: The smallest increment an instrument can measure (its resolution or least count). A ruler marked in millimeters cannot reliably measure to tenths of a millimeter.

Common Mistakes When Calculating Percent Error

Mistake 1: Forgetting the absolute value

The most common computational error. Without absolute value, you can get a negative percent error, which is generally not the standard academic format. Always apply |absolute value| to the numerator, or to the entire expression before multiplying by 100.

Wrong: (47 − 50) ÷ 50 × 100 = −6% Right: |47 − 50| ÷ 50 × 100 = 6%

Mistake 2: Using the experimental value as the denominator

The denominator must always be the theoretical (true) value, not the experimental value. Dividing by the experimental value gives a different calculation called the "percent difference" between two values — not percent error.

Wrong: |47 − 50| ÷ 47 × 100 = 6.38% Right: |47 − 50| ÷ 50 × 100 = 6.00%

Mistake 3: Confusing percent error with percent difference

Percent error requires a known true value. Percent difference compares two measured values when neither is definitively "true." They use different denominators:

• Percent Error denominator: the theoretical/true value
• Percent Difference denominator: the average of the two values: (Value1 + Value2) ÷ 2

Mistake 4: Using inconsistent units

Experimental and theoretical values must be in the same units before calculating. If the theoretical value is in grams and the experimental value is in milligrams, convert first.

Mistake 5: Reporting too many decimal places

A percent error of 4.182763% implies a level of precision that most experimental measurements cannot support. Report to 2–3 significant figures in most cases.

Mistake 6: Not squaring percent errors when combining

When averaging multiple percent errors, you do not simply average them arithmetically. Root Mean Square (RMS) error is the more statistically appropriate method: √(average of squared errors). For most student lab work, arithmetic averaging is acceptable — but note the distinction for advanced work.

Related Math and Science Tools

Percent error sits within a broader ecosystem of mathematical and statistical analysis tools. These related calculators on WithinSecs complete your scientific and academic toolkit:

For General Math:

• Percentage Calculator — All types of percentage problems including percent change, percent increase, and percent decrease
• Scientific Calculator — Full-precision arithmetic for complex scientific calculations
• Algebra Calculator — Solve algebraic equations that arise in physics and chemistry problems
• Exponent Calculator — Powers and scientific notation for very large or very small values

For Statistics and Data Analysis:

• Standard Deviation Calculator — Measure precision and variability of repeated measurements
• Average Calculator — Mean of multiple experimental trials
• Median Calculator — Robust central tendency for datasets with outliers
• Equation Solver Calculator — Solve equations derived from experimental data

For Unit Conversion in Lab Work:

• Unit Converter — Convert between metric and imperial units before calculating error
• Grams to Ounces — Convert mass units for chemistry experiments
• Milliliter to Ounce — Convert volume units for liquid measurements

Conclusion: Master Percent Error WithinSecs

Percent error is the universal language of measurement accuracy. Whether you are verifying a chemistry titration, checking a physics experiment against theoretical prediction, assessing manufacturing quality, or evaluating a business forecast, the same formula applies: take the absolute difference, divide by the true value, multiply by 100.

The formula is straightforward. Applying it correctly — using the right denominator, applying absolute value, understanding when to report signed error, and interpreting the result in context — is where genuine understanding separates good students and professionals from those who merely know the equation.

The WithinSecs Percent Error Calculator gives you instant results with the full formula shown — so you are learning the method, not just getting an answer. Enter your experimental value, your theoretical value, and get your percent error with complete working in seconds.

Bookmark this page. Share it with your classmates, lab partners, and study group. And explore the full WithinSecs math toolkit — from the Percentage Calculator and Standard Deviation Calculator to the Scientific Calculator — for every calculation from first-year chemistry through advanced research.

Every formula. Every step. Every answer. WithinSecs.